# A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare; Shuntaro Yamagishi

Acta Arithmetica (2014)

- Volume: 166, Issue: 1, page 55-67
- ISSN: 0065-1036

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topKathryn E. Hare, and Shuntaro Yamagishi. "A generalization of a theorem of Erdős-Rényi to m-fold sums and differences." Acta Arithmetica 166.1 (2014): 55-67. <http://eudml.org/doc/279059>.

@article{KathrynE2014,

abstract = {Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_\{N\}^\{(m)\}(ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_\{N\}^\{(m)\}(ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.},

author = {Kathryn E. Hare, Shuntaro Yamagishi},

journal = {Acta Arithmetica},

keywords = {sequences; additive number theory; probabilistic methods; thin sets},

language = {eng},

number = {1},

pages = {55-67},

title = {A generalization of a theorem of Erdős-Rényi to m-fold sums and differences},

url = {http://eudml.org/doc/279059},

volume = {166},

year = {2014},

}

TY - JOUR

AU - Kathryn E. Hare

AU - Shuntaro Yamagishi

TI - A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

JO - Acta Arithmetica

PY - 2014

VL - 166

IS - 1

SP - 55

EP - 67

AB - Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)}(ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)}(ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

LA - eng

KW - sequences; additive number theory; probabilistic methods; thin sets

UR - http://eudml.org/doc/279059

ER -

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